can be written in terms of two invariant functions, \(\Pi _{\perp \perp }(z)\)  and \(\Pi _{\parallel \perp }(z)\) , which correspond to contributions with different Lorentz symmetry and do not mix under renormalization [1]}, as
\(\Pi _{\mu \nu \alpha \beta }(z) &=&(g^\perp _{\mu \alpha }g^\perp _{\nu \beta } - g^\perp _{\nu \alpha }g^\perp _{\mu \beta }) \Pi _{\perp \perp }(z)+ (g^\parallel _{\mu \alpha } g^\perp _{\nu \beta } - g^\parallel _{\nu \alpha } g^\perp _{\mu \beta }- g^\parallel _{\mu \beta } g^\perp _{\nu \alpha } + g^\parallel _{\nu \beta } g^\perp _{\mu \alpha })\Pi _{\parallel \perp }(z)\nonumber \\&=&(g_{\mu \alpha }g_{\nu \beta } - g_{\nu \alpha }g_{\mu \beta })\, \Pi _{\perp \perp }(z)\nonumber \\&&{}+ \frac{1}{v^2}( v_\mu v_\alpha g_{\nu \beta } - v_\nu v_\alpha g_{\mu \beta }-v_\mu v_\beta g_{\nu \alpha } + v_\nu v_\beta g_{\mu \alpha })\, [\Pi _{\parallel \perp }(z) - \Pi _{\perp \perp }(z)]\,,\) 
